Recently, Greenfeld and Tao disprove the conjecture that translational tilings of a single tile can always be periodic [Ann. Math. 200(2024), 301-363]. In another paper [to appear in J. Eur. Math. Soc.], they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $\mathbb{Z}^n$ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of $\mathbb{Z}^n$ with a monotile is undecidable, for some fixed $n$. This paper shows that translational tiling of the $3$-dimensional space with a set of $5$ polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from $3$-dimensional space to $4$-dimensional space, we manage to show that translational tiling of the $4$-dimensional space with a set of $4$ tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the $n$-dimensional space with a monotile, for some fixed $n$.

翻译：最近，Greenfeld 与 Tao 否定了关于单块铺砌体的平移铺砌必然具有周期性的猜想 [Ann. Math. 200(2024), 301-363]。在另一篇论文 [即将发表于 J. Eur. Math. Soc.] 中，他们还证明了当维度 $n$ 作为输入的一部分时，使用单块铺砌体对 $\mathbb{Z}^n$ 子集的平移铺砌问题是不可判定的。这两项结果为以下猜想提供了强有力的证据：对于某个固定的 $n$，使用单块铺砌体对 $\mathbb{Z}^n$ 的平移铺砌是不可判定的。本文证明了使用一组 $5$ 块多立方体对三维空间的平移铺砌是不可判定的。通过引入一种将一组多立方体及其铺砌从三维空间提升到四维空间的技术，我们成功证明了使用一组 $4$ 块铺砌体对四维空间的平移铺砌是不可判定的。这是朝着解决“对于某个固定的 $n$，使用单块铺砌体对 $n$ 维空间的平移铺砌具有不可判定性”这一猜想所迈出的一步。